Teaching GuideTerm
Higher Technical University College of Nautical Science and Naval Engines
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Grao en Náutica e Transporte Marítimo
 Subjects
  Mathematics I
   Contents
Topic Sub-topic
Lesson 1.- Espazos Vectoriais
 1.1.- Vector space. Definition. Examples and Properties
1.2.- Vector subspace.
1.3.- System of Generators of a Subspace
1.4.- Linear Independence
1.5.- Basis of a Vector Space. Finite Dimensional Spaces.
1.6.- Change of Basis in a Vector Space
1.7.- Union and Intersection of Subspaces
1.8.- Sum of Subspaces. Direct sum. Supplementary Subspaces.
1.9.- Product of Vectorial Spaces
Lesson 2.- Linear Functions. Matrices. 2.1.- Linear Function: Definition, Examples, Properties and Types of Linear Functions.
2.2.- Kernel and Image of a Linear Function.
2.3.- Existence and obtention of an Associated Matrix to a Linear Function.
2.4.- Addition of Linear Functions. Product by a Scalar. Associated Matrices.
2.5.- Vector Spaces of Matrices
2.6.- Composition of Linear Functions. Associated Matrix.
2.7.- Product of Matrices. Ring of Square Matrices
2.8.- Some Particular Types of Matrices
2.9.- Transpose Matrix. Symmetric, Antisymmetric and Orthogonal Matrices.
2.10.- Matrices of Complex Elements.
Lesson 3.- Determinants.
 3.0.- Permutations. Class of a Permutation.
3.1.- Determinant of a Square Matrix. Sarrus Rule.
3.2.- Properties of Determinants.
3.3.- Methods for Calculation of Determinants. Cofactor Matrix.
3.4.- Product of Determinants.
3.5.- Some Particular Examples of Determinants.
3.6.- Reverse Matrix.
3.7.- Rank of a Matrix.
3.8.- Rank of a System of Vectors
3.9.- Expression of the Change of Base of a Vectorial Space in shape Matrix
Lesson 4.- Systems of Linear Equations.
 4.1.- Definitions. Classification. Matrix notation.
4.2.- Equivalent systems.
4.3.- System of Cramer. Rule of Cramer
4.4.- General System of Linear Equations. Theorem of Rouché-Frobenius
4.5.- Homogeneous Systems.
4.6.- Methods of Resolution by Reduction. Gauss' Method.

Lesson 5.- Matrix Diagonalization. 5.1.- Eigenvectors and Eigenvalues. Properties.
5.2.- Characteristic polynomial. Properties.
5.3.- Diagonalizable Matrices. Diagonalization.
5.4.- Diagonalization Of Symmetric Matrices.

Lesson 6.- Affine Space E3. Problems of Incidence and Parallelism.
 6.1.- Affine Space Associated to a Vector Space. System of Reference. Coordinates.
6.2.- Equations of Straight Lines.
6.3.- Relative positions of Straight Lines.
6.4.- Equations of a Plane.
6.5.- Relative positions of Planes. Bundles of Planes.
6.6.- Relative positions of Straight Lines and Planes.

Lesson 7.- Euclidean Vector Spaces. Scalar product, Vector product. Mixed Product. 7.1.- Scalar product
7.2.- Determination of a Scalar Product. Gram Matrix.
7.3.- Euclidean Vector Space. Definition.
7.4.- Norm of a Vector. Relevant Equalities and Inequalities.
7.5.- Angle of two Vectors. Orthogonality.
7.6.- Orthonormal Basis. Expression of the Scalar Product in an Orthonormal Basis.
7.7.- Euclidean Space E3.
7.8.- Orientation in E3.
7.9.- Vector product in R3 . Properties. Analytical expression.
7.10.- Mixed product. Analytical expression. Geometrical interpretation.
7.11.- Combined Products.
Lesson 8.- Metric Problems in Euclidean Spaces.
 8.1.- Normal equation of a Plane.
8.2.- Angles between Linear Manifolds in R3: Angle of Two Planes, Angle of Two Straight Lines, Angle of Straight Line and Plane.
8.3.- Distance between Linear Manifolds in R3: Distance of a Point to a Plane, Distance of a Point to a Straight Line. Distance between two Planes, Distance between Straight Line and Plane. Distance between two Straight Lines. Common Perpendicular to two Straight Lines.
8.4.- Cylindrical coordinates and Spherical coordinates in R3.

Lesson 9.-Real valued functions of a Real Variable. Continuity.
 9.1.- Basic definitions.
9.2.- Functional limits.
9.3.- Continuity. Types of Discontinuity.
9.4.- Properties and Theorems on Continuous Functions.

Lesson 10.- Differentiability and Applications of the Derivatives.
 10.1.- Derivative and Differential of a Function in a Point. Geometrical meaning.
10.2.- Properties and Calculation of Derivatives.
10.3.- Derivative function. Successive derivatives.
10.4.- Applications of the Derivatives to the Local Study of a Function: Growth and Decreasing. Maxima and Minima. Concavity and Convexity. Inflection points.
10.5.- Theorems of Rolle and Mean Value Theorem.
10.6.- Rules of L´Hôpital

Lesson 11.- Theorem of Taylor. Applications.


 11.1.- Expression of a Polynomial by means of his Derivatives in a Point.
11.2.- Polynomial and Theorem of Taylor. Formulae of Taylor and Mac Laurin.
11.3.- Expression of Lagrange for the Residual. Bounds for the residual.
11.4.- Applications to the Local Study of a Function: Monotonicity. Extremal values. Concavity and Convexity. Inflection points.
Lesson 12.- Graphic representation of Real Valued Functions.
12.1.- Domain and Continuity
12.2.- Symmetries
12.3.- Periodicity.
12.4.- Intersection with the coordinates axis.
12.5.- Use of successive derivatives and applications: Monotonicity. Extremal values. Concavity and Convexity. Inflection points.
12.6.- Asymptotes and Parabolic Branches

Lesson 13.- Sequences and Series.
 13.1.- General definitions. Types of Sequences.
13.2.- Practical calculation of Limits
13.3.- General definitions. Main Types of Numerical Series.
13.4.- Properties of the Numerical Series. Criteria of Convergence for Series of Positive Terms.
13.5.- Series of Positive and Negative Terms. Alternated Series.

Lesson 14.- Functional Sequences and Series. Series of powers. 14.1.- General definitions.
14.2.- Series of Powers. Convergence.
14.3.- Series expansions.
14.4.- Series of Taylor and Mac Laurin.
14.5.- Binomial Series.
14.6.- Method of the Undetermined Coefficients.

Lesson 15.- Indefinite integration of Functions of a Real Variable 15.1.- General definitions. Table of Primitives.
15.2.- Immediate integration
15.3.- Integration by Parts
15.4.- Integration of Rational Functions
15.5.- Integration by Replacement or Change of Variable
Lesson 16.- Definite Integration. Applications. 16.1.- General definitions
16.2.- Properties
16.3.- Mean Value Theorem. Barrow's Rule.
16.4.- Evaluation of Definite Integrals.
16.5.- Improper Integral.
16.6.- Applications of the Definite Integral

Lesson 17.- Complex Numbers 17.1.- General definitions
17.2.- Fundamental operations
17.3.- Powers and Roots
17.4.- Exponential form of a Complex
17.5.- Logarithms And Complex Powers.
The development and overcoming of these contents, together with those corresponding to other subjects that include the acquisition of specific competencies of the degree, guarantees the knowledge, comprehension and sufficiency of the competencies contained in Table AII / 2, of the STCW Convention, related to the level of management of chief mates of the Merchant Navy, on ships without gross tonnage limitation and Master up to a maximum of 500 GT. Table A-II / 2 of the STCW Convention.

Mandatory minimum requirements for certification of masters and chief mates on chief on ships of 500 gross tonnage or more.
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